Determinant Evaluation Methods: Up to 3x3 Order

Determinant Evaluation Up to 3x3 Order

The determinant of a matrix is a scalar value used to describe its orientation and volume properties. It can be computed using several methods, including the expansion by minors along rows and columns. In this article, we will discuss the determinant evaluation up to 3x3 order, with a focus on the expansion method.

Expansion Method

First Order Determinants

In the first order case, there is only one minor and it is equal to the element itself. Thus, the determinant evaluates directly from the elements of the row or column chosen for expansion. For example, if we consider the matrix A = |a b c| and expand by minors along row 1, the determinant D(A) is calculated as follows:

D(A) = a * a - b * b + c * c

This results in a determinant of a^2 - b^2 + c^2. If we choose to expand along row 2 instead, we get:

D(A) = -a * b + b * c - c * a

This also results in the same determinant of a^2 - b^2 + c^2.

Second Order Determinants

In the second order case, we have two minors: one formed by selecting the first two columns and the second formed by selecting the last two columns. When expanding by minors along row 1, we get:

D(A) = (a * d - b * c) - (b * e - c * d)

D(A) = a * d - b * e - (b * e - c * d)

D(A) = a * d - b * e - b * e + c * d

D(A) = (a - b) * d + c * e

Similarly, when expanding by minors along row 2, we get:

D(A) = (b * f - c * e) - (a * f - b * e)

D(A) = b * f - c * e - a * f + b * e

D(A) = (b - a) * f + e * e

D(A) = (b - a) * f + e^2

Third Order Determinants

In the third order case, we have three minors formed by selecting the first three columns, the middle three columns, and the last three columns. When expanding by minors along row 1, we get:

D(A) = (a * g - b * f) - (b * h - c * g) + (b * h - c * i)

D(A) = a * g - b * f - b * h + c * g + b * h - c * i

D(A) = (a - b) * g - b * f - b * h + (b - c) * g + b * h - (b - c) * i

D(A) = (a - b) * g - b * f - b * h + (b - c) * g - b * h + (b - c) * i

D(A) = (a - b) * g - b * f + b * i - (b - c) * h

Generalization

The expansion method can be generalized to any order of the determinant. The determinant of a square matrix M of order n can be calculated using the following formula:

D(M) = ∑(-1)^(i+j) * M_ij * minors of M

Here, the sum is taken over all possible choices of i and j, where i and j are the row and column indices, respectively, and M_ij is the element in the i-th row and j-th column of the matrix M. The minors of M are the determinants of the submatrices obtained by deleting i-th row and j-th column from the matrix M.